On the smoothness of positive definite and radial functions

We consider positivedefinite and radialfunctions. After giving general results concerning the smoothness of general positivedefinite and radialfunctions, we investigate the class of compactly supported, positivedefinite, and radialfunctions, where every function consists of a univariate polynomial within its support. Especially, we show that these functions necessarily possess an even number of continuous derivatives. Finally, we provide a general construction technique which we use to construct a new family of compactly supported basis functions of arbitrary smoothness.